9 MEASURABLE FUNCTIONS 149 (c) Take X to be the circle U : z\ = 1 in C, and ju, to be the image of Lebesgue measure under the mapping t\-+e2nit of [0, 1] onto U. If 6 is irrational, show that the mapping u : z\-+e2ntez leaves ft invariant and is ergodic with respect to yi. (Use BohPs theorem, Section 13.4, Problem 7.) (d) Suppose that X is compact and u ergodic with respect to jit. If A and B are jLt-measurable subsets of X, show that lim - I]' p,(u~k(A) n B) = jLc(A)/x(B)/ju(X). n-*oo n k= 0 Conversely, if this formula is satisfied for all pairs of measurable subsets A, B in X, then u is ergodic with respect to p. (e) Suppose that X is compact and u ergodic with respect to /x. If a measurable function/^ 0 is such that lim -x;1 /<>*(*)) n-*oo ft fc = 0 exists almost everywhere, then /is integrable. (Consider /as the limit of an increasing sequence of bounded functions.) (f) Suppose that X is compact, that u is ergodic with respect to /x, and that Supp(ju) = X. Show that, for almost all # <= X, the set of points if(x) (n ^> 0) is dense inX. 14. Let X be a compact space, u : X -> X a continuous mapping, and p, a positive measure on X, invariant with respect to u and of total mass 1 . Let A be a jLt-measurable subset of X. For each x e A, let n(x) denote the smallest integer n J> 1 such that un(x) e A. If we put An = u~n(A) and Bn = w ~"(X — A), then the set of points x e A for which n(x) — m is A0 n Bj n B2 r\ • - • n Bm_] n Am, and hence the function x\—*n(x) is measurable. (a) Put a0 = 1 and aw = /x(B0 n BI n • • • n Bm.~i) (m ^> 1). Show that BI n B2 n • • • n Bm_i n Bm n Am + i) = am — 2am+1 -j- for all m ;> 0. (b) Show that the series whose general term is am — am+1 is convergent and that its terms form a decreasing sequence (interpret ocm — am+i as the measure of a set). Deduce that lim m(ocm — am + 1) == 0. (c) Let Bo, = fl B" • Snow tnat ' n(x) C'Kac's theorem": use (a) and (b)). Consider the case where u is ergodic with respect to ju, and /x(A) > 0. (d) Suppose that u is bijective and ergodic with respect to /x, and that jtt(A) > 0. Let Em be the set of points x e A such that n(x) = m. Show that the complement in A of 0 Em is negligible (cf. Problem 11 (a)). Show that the sets up(Em) for m ^ 1 m- 1 and 0 <; p < m are mutually disjoint, and that the complement of their union in X is negligible ("Kakutani's skyscraper")-consider first the case where/is bounded; then pass to the general case by observing to show that it implies that there exists a rational number r